Page 599 - Aircraft Stuctures for Engineering Student

580 Elementary aeroelasticity

P.13.7 Figure P.13.7 shows the idealized cross-section of a single cell tube with
axis of symmetry xx and length 1525 mm in which the direct stresses due to bending
are carried only in the four booms of the cross-section. The walls are assumed to carry
only shear stresses. The tube is built-in at the root and carries a weight of 4450 N at its
tip; the centre of gravity of the weight coincides with the shear centre of the tube cross-
section. Assuming that the direct and shear stresses in the tube are given by basic
bending theory, calculate the natural frequency of flexural vibrations of the weight
in a vertical direction. The effect of the weight of the tube is to be neglected and it
should be noted that it is not necessary to know the position of the shear centre of
the cross-section. The effect on the deflections of the shear strains in the tube walls
must be included

E = 70000N/mm2, G = 26 500N/mm2, boom areas 970mm2
Ans. 12.1 Hz.

-I
I 525 rnrn

1.25 rnrn

1.0 mrn
I.Ornrn

600 mm

Fig. P.13.7

P.13.8 A straight beam of length 1 is rigidly built-in at its ends. For one quarter of
its length from each end the bending stiffness is 4E1 and the mass/unit length is 2m:
for the central half the stiffness is EI and the mass m per unit length. In addition, the
beam carries three mass concentrations, $ml at $1 from each end and am1 at the
centre, as shown in Fig. P.13.8.
Use an energy method or other approximation to estimate the lowest frequency
of natural flexural vibration. A first approximation solution will suffice if it is
accompanied by a brief explanation of a method of obtaining improved accuracy.

Ans. 3.7g